Artículos

Construction of Rotation Symmetric Boolean Functions with optimal Algebraic Immunity*

Construcción de Funciones Booleanas de Rotación Simétrica con Inmunidad Algebraica Óptima

Sumanta Sarkar¹ and Subhamoy Maitra²

¹ SECRET INRIA Rocquencourt, B.P. 105 78153 Le Chesnay Cedex, FRANCE. E–mail: sumanta.sarkar@inria.fr

² Applied Statistics Unit, Indian Statistical Institute, 203, B T Road, Calcutta 700 108, INDIA. E–mail: subho@isical.ac.in

Article received on March 1, 2008
Accepted on June 14, 2008

Abstract

In this paper, we present theoretical constructions of Rotation Symmetric Boolean Functions (RSBFs) on odd number of variables with the maximum possible algebraic immunity. To get high nonlinearity, we generalize our construction to a search technique in the RSBF class. We present RSBFs with the maximum algebraic immunity and high nonlinearity for odd number of variables. We also study the RSBFs on even number of variables for maximum algebraic immunity.

Keywords: Algebraic Immunity, Boolean Function, Nonlinearity, Nonsingular Matrix, Rotational Symmetry, Walsh Spectrum.

Resumen

En este artículo, presentamos construcciones teóricas de funciones Booleanas de rotación simétrica (RSBFs por sus siglas en inglés) con un número impar de variables y con máxima inmunidad algebraica. Con el objeto de obtener funciones Booleanas de muy alta no linealidad, generalizamos nuestra construcción a una técnica de búsqueda en la clase RSBF. Presentamos así RSBFs con inmunidad algebraica máxima y alta no linealidad para un número impar de variables, y también RSBFs con un número par de variables que exhiben inmunidad algebraica máxima.

Palabras Claves: Inmunidad algebraica, funciones Booleanas, no–linealidad, matrices no singulares, simetría rotacional, Espectro de Walsh.

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Acknowledgments

The authors would like to thank the anonymous reviewer for his comments and suggestions on this paper.

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Note

* This in an extended and revised version of the paper (Sarkar and Maitra 2007).